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<b>let </b><font color="Maroon">q</font> be   <a href="rat_1.html#NM1">Rational</a>;<br/>

<b>set </b><font color="Maroon">k</font> =  <a href="rat_1.html#K2">numerator</a> <font color="Maroon">q</font>;<br/>
<b>set </b><font color="Maroon">h</font> =  <a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font>;<br/>
<a NAME="E1:59"/><i><font color="Green">E40</font></i>: 
 <a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="rat_1.html#D3">RAT_1:def 3</a></i>;<br/>
<b>assume </b><a NAME="E2:59"/><i><font color="Green">E41</font></i>: 
not  <a href="rat_1.html#K2">numerator</a> <font color="Maroon">q</font>, <a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font> <a href="int_2.html#R1">are_relative_prime</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">a</font> = <span class="p1">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">q</font></span>)</span> <a href="int_2.html#K3">gcd</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_2.html#T29">INT_2:29</a></i>;<br/>
<a NAME="E4:59"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E7"><i><font color="Green">Lemma44</font></i></a>, <a class="ref" href="int_2.html#D4">INT_2:def 4</a></i>;<br/>
<a NAME="E5:59"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="wsierp_1.html#T5" target="_self">Th5</a>, <a class="ref" href="int_2.html#T35">INT_2:35</a></i>;<br/>
<a NAME="E6:59"/><b>then </b>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E7:59"/><b>then </b>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E8:59"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E8"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E9:59"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">a</font> <a href="int_1.html#R2">divides</a>  <a href="rat_1.html#K2">numerator</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T31">INT_2:31</a></i>;<br/>
<a NAME="E10:59"/>
<span class="p1">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">q</font></span>)</span> <a href="int_2.html#K3">gcd</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font></span>)</span> <a href="int_1.html#R2">divides</a>  <a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T32">INT_2:32</a></i>;<br/>
<a NAME="E11:59"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">a</font> <a href="nat_1.html#R1">divides</a>  <a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">l</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E13:59"/><i><font color="Green">E105</font></i>: 
 <a href="rat_1.html#K2">numerator</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E12"><i><font color="Green">Lemma47</font></i></a>, <a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E15:59"/><i><font color="Green">E106</font></i>: 
 <a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E4:59"><i><font color="Green">E48</font></i></a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E17:59"/>
 <a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E9:59"><i><font color="Green">E50</font></i></a></i>;<br/>
<b>hence </b><a NAME="E18:59"/>
contradiction
 <b>by </b><i><a class="txt" href="wsierp_1.html#E9"><i><font color="Green">Lemma46</font></i></a>, <a class="txt" href="wsierp_1.html#E8:59"><i><font color="Green">E49</font></i></a>, <a class="ref" href="rat_1.html#T62">RAT_1:62</a></i>;<br/>


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